Problem 114 - Elementary Algebra

Question

Simplify. $$ (12) 2-34 \div 916-13 $$

Step-by-Step Solution

Answer

The simplified result is approximately 10.96.

Step 1: Evaluate Multiplication

First, evaluate the multiplication part of the expression. We have \( (12) \times 2 \). Calculating this gives us \( 24 \). Now the expression becomes: \[ 24 - 34 \div 916 - 13 \]

Step 2: Evaluate Division

Next, we perform the division. We have \( 34 \div 916 \). Since \( 34 \div 916 \approx 0.0371 \) which is a small value, we can keep it as a decimal for now. The expression now reads: \[ 24 - 0.0371 - 13 \]

Step 3: Perform Subtractions

Now, perform the subtractions in order from left to right. Start with \( 24 - 0.0371 \), which equals \( 23.9629 \). Then, subtract \( 13 \) from \( 23.9629 \):\( 23.9629 - 13 \approx 10.9629 \).

Step 4: Round the Result

The final result is \( 10.9629 \). You can round this to two decimal places, giving \( 10.96 \).

Key Concepts

Order of Operations

When you simplify an expression like the one given, you need to follow the order of operations, often remembered by the acronym PEMDAS:

• Parentheses first.
• Exponents (ie. powers and square roots, etc.)
• Multiplication and Division (left-to-right)
• Addition and Subtraction (left-to-right)

This means that multiplication and division should be handled before addition and subtraction. In our exercise, the multiplication is done first, followed by division, and finally, subtraction steps.

This systematic approach ensures accuracy and prevents mistakes. Ignoring this order can lead to incorrect results. For instance, doing the subtraction before division or multiplication would give a completely different outcome.

Decimal Approximation

In the exercise, we come across a division that results in a non-integer, specifically \( 34 \div 916 \approx 0.0371 \). When the result of a division isn't a whole number, you often work with decimal approximations.

Using decimal approximations helps simplify arithmetic tasks, especially when the exact decimal has many digits or when the number goes on infinitely, known as a repeating decimal. Here, we rounded \( 0.0371 \) sufficiently because it is practical and does not substantially affect the overall result.

You can control the decimal places to which you round, depending on the precision needed for your specific task. Often, rounding to two or three decimal places is enough for most calculations in homework settings. This makes it easier to read and compute further operations.

Arithmetic Operations

After understanding the order of operations and handling decimals, it’s essential to apply arithmetic operations correctly. Arithmetic operations (addition, subtraction, multiplication, and division) are the foundation of algebra.

• Multiplication: Do this first as per order rules, like \( 12 \times 2 \) to get \( 24 \).
• Division: Followed by division, turning \( 34 \div 916 \) into \( 0.0371 \).
• Subtraction: Finally, subtract step by step, from left to right with \( 24 - 0.0371 \) and then subtract \( 13 \) from the result.

Each step builds on the previous result. Precision in each operation is key, especially when dealing with decimals. Arithmetic operations are straightforward, but ensuring you follow the correct operations in sequence is crucial to achieving the right solution.